Algebra and Applications 2. Группа авторов

which is symmetric in t₁ and t₂ thanks to the left pre-Lie relation in A. The expression [1.81] is then the same if we exchange t₁ with the branch t₂ or any other branch; hence, it is invariant by any permutation of the branches t₁,…, t_n. This proves Theorem 1.4 for one generator. The general case is proven similarly, except we have to replace a ∈ A by a collection {a₁,…,ad}.

1.6.3. Right-sided commutative Hopf algebras and the Loday-Ronco theorem

      Loday and Ronco (2010) have found a deep link between pre-Lie algebras and commutative Hopf algebras of a certain type: let ℋ be a commutative Hopf algebra. Following this, we say that ℋ is right-sided if it is free as a commutative algebra, that is, ℋ = S(V) for some k-vector space V, and if the reduced coproduct verifies:

      [1.83]

      Suppose that V = ⊕_{n ≥ 0} is graded with finite-dimensional homogeneous components. Then, the graded dual A = V⁰ is a left pre-Lie algebra, and by the Milnor-Moore theorem, the graded dual ℋ⁰ is isomorphic to the enveloping algebra as graded Hopf algebra. Conversely, for any graded pre-Lie algebra A, the graded dual is free commutative right-sided (Loday and Ronco 2010, Theorem 5.3).

      The Hopf algebra ℋ_CK of rooted forests enters into this framework, and, as it was first explicited in Chapoton (2001), the associated pre-Lie algebra is the free pre-Lie algebra of rooted trees with grafting: to see this, denote by (δs) the dual basis in the graded dual of the forest basis of ℋ_CK. The correspondence δ : s ↦ δs extends linearly to a unique vector space isomorphism from ℋ_CK onto . For any tree t, the corresponding δt is an infinitesimal character of ℋ_CK, that is, it is a primitive element of ℋ°. We denote by * the (convolution) product of ℋ°. We have:

      [1.84]

      Here, t ↷ u is obtained by grafting t on u, namely:

      [1.85]

      where N′(t, u, v) is the number of partitions V(t) = V ∐ W, W < V, such that υ|_V = t and υ|_W = u. Another normalization is often employed: considering the normalized dual basis , where σ(t) = |Aut t| stands for the symmetry factor of t, we obviously have:

      [1.86]

where:

      [1.87]

where can be interpreted as the number of ways to graft the tree t on the tree u in order to get the tree v. The operation → then coincides with the grafting free pre-Lie operation introduced in section 1.6.1.2².

      The other pre-Lie operation ⊲ of section 1.6.1.2, more precisely its opposite ⊳, is associated with another right-sided Hopf algebra of forests ℋ which has been investigated in Calaque et al. (2011) and Manchon and Saidi (2011), and which can be defined by considering trees as Feynman diagrams (without loops): let be the vector space spanned by rooted trees with at least one edge. Consider the symmetric algebra , which can be seen as the k-vector space generated by rooted forests with all connected components containing at least one edge. We identify the unit of with the rooted tree •. A subforest of a tree t is either the trivial forest •, or a collection (t₁,…, tn) of pairwise disjoint subtrees of t, each of them containing at least one edge. In particular, two subtrees of a subforest cannot have any common vertex.